Airbus A380 Air France from CDG to LAX got some flaps issue when arriving on final @ LAX. I tried to sum up the conversation between ATC & pilot. Then he switched to 134.35 , apparently this frequency is not feeded on LiveATC. Landed 30 minutes later without additionnal issue.

You declare a pan pan pan once. Afaik you don't add it to your call sign every time you use the radio.

At least when declaring a mayday, I was indeed taught to add "mayday" to my callsign. It keeps reminding everyone that an exceptional situation is still going on. The reminder may not be necessary for the controller, though it doesn't harm, especially when a situation takes very long (flying holding patterns to burn off fuel), or when you switch frequencies (approach to tower).

But I think the main reason is to raise other pilots' awareness. And new pilots are joining the frequency all the time - they don't know what was declared before, neither which aircraft is affected. But it's better if everyone knows that there is an emergency in progress and who is affected (make extra sure today, that you keep your transmissions really short, don't cause delays etc).

According to the ICAO phraseology, AIM, and pilot/controller glossary, Pan-Pan, Pan-Pan, Pan-Pan is a prefix used to denote an urgency message. I have seen no source that states that one should add pan-pan to one's callsign, even for non-urgency messages. Arguably the initial call about flaps not working should have been a pan-pan. The subsequent messages about holding fixes and direct VORs did not need a pan. Here's another example straight out of the relevant ICAO guide, example of Emergency Communications:

As you can see, Mayday Mayday does not become part of the call sign. It is used for the initial call-up to denote a distress call or distress situation, but each message after that is not itself a distress call. Same with Pan pan.

Likewise, the AIM states that the initial communication should be prefixed with Mayday x 3 or Pan-Pan x 3. It does not state that the mayday or pan-pan becomes part of the callsign.

Not a big deal, but the AF pilot just sounds kind of ridiculous. Every time he spoke I rolled my eyes. But the French have that effect on me in general.

As soon as there is any doubt as to the safe conduct of a flight, immediatelyrequest assistance from ATC. Flight crews should declare the situation early; itcan always be cancelled.!A distress call (situation where the aircraft requires immediate assistance) isprefixed: MAYDAY, MAYDAY, MAYDAY.!An urgency message (situation not requiring immediate assistance) isprefixed: PAN PAN, PAN PAN, PAN PAN.!Make the initial call on the frequency in use, but if that is notpossible squawk 7700 and call on 121.5.!The distress/urgency message shall contain (at least) the name of thestation addressed, the callsign, nature of the emergency, fuel enduranceand persons on board; and any supporting information such as position,level, (descending), speed and heading, and pilot’s intentions."